Talk:Simply connected

The article said:

A subset X of Euclidean 2-space R² is simply connected if and only if both X and R² - X are connected.

I tried to correct this yesterday by changing "subset" to "bounded subset" and "connected" to "path-connected", but it's still wrong. It is possible to construct a bounded simply connected subset of the plane with disconnected complement. --Zundark, 2002 Feb 19

Thanks for finding the bug. I mindlessly copied it from Mathworld. Maybe we need X to be open? AxelBoldt

Never trust Mathworld. I think

A bounded open subset X of Euclidean 2-space R² is simply connected if and only if both X and R² - X are connected.

may be correct, but we need to check it before adding it to the article. --Zundark, 2002 Feb 19

a hollow ball is?

I smell inconsistency: a hollow ball (in 3D) can't be simply connected any more than a circle (in 2D). Whether or not they have a hole depends on your definition of hole.

Yes, it depends on your definition of a hole, which is why we give a formal of definition of simply connected which is unambiguous. A hollow ball (a sphere) is simply connected - it's intuitively obvious that a circle on a sphere can be contracted to a point. --Zundark 15:52 Nov 28, 2002 (UTC)

(I'm here to look up these concepts because the descriptions are infinitely better than those in the books of our math library's geometry/topology section, but there seems to be an imperfection here.)

The definition given is over my head: why involve the circle? If "simply connected" is meant to express the notion "completely filled up", isn't it better to define it as a connected subspace of which the complement's interior is also connected, or something like that?

Your definition does not give an intrinsic property of the space, it gives a property of the embedding of the space in some larger space. The idea of "simply connected" is that there is a path from every point to every other point (this is path-connectedness) and that there is essentially only one such path (thus "simply" connected), in the sense that any two paths from A to B can be deformed into one another. --Zundark 15:52 Nov 28, 2002 (UTC)